

where \(j\)  is summed over the top \((t)\)  and bottom \((b)\)  layers, \(S\) ,\(S^{\prime }\)  are summed over the the \(A\)  and \(B\)  sublattices, and \(\mu \)  over \(x\)  and \(y\)  components.
In the above, we split \(H_{intra}\)  into two terms: \(H^{(0)}_{intra}\)  and \(\delta H_{intra}\) . The first term, \(H^{(0)}_{intra}\) , contains all the contributions up to the second order in gradients whose energy scale is above 1meV and dominates the second term \(\delta H_{intra}\) . The numerical values of the coefficients appearing in \(H^{(0)}_{intra}\)  for the two different microscopic models [1]}, [2]} can be found in the Table. REF . The definition of the lattice distortion induced pseudo vector fields \(\mathcal {A}(x)\)  and the fields \(\xi _{SS^{\prime }}(x)\)  are discussed below. The second term, \(\delta H_{intra}\) , contains other second order gradient contributions that are smaller than 1meV. We have checked numerically that inclusion of \(\delta H_{intra}\)  does not improve the agreement between the spectra of the tight binding Hamiltonians and \(H_{eff}\)  in any significant way, as demonstrated in the Fig. REF  where \(\delta H_{intra}\)  is omitted. Therefore, we will focus on \(H^{(0)}_{intra}\)  in the main text. For completeness, we spell out the details of \(\delta H_{intra}\)  in the appendix Sec. .
